Quantitative Aptitude PYQs

Shortcut Trick

Efficiency ratio A : B = 4 : 5. Since time is inversely proportional to efficiency, time ratio A : B = 5 : 4.

Difference in time ratio parts = 1 unit. Actual difference = (x + 5) − (x + 2) = 3 days.

Total Work = 4 × (15 + 5) = 80 units. In 2 days (B then A), work done = 5 + 4 = 9 units.

In 12 days (6 cycles), work = 9 × 6 = 54 units. Error in ratio assumption; let's use x calculation.

∴ The correct answer is 13.25 days.

Alternate Method

Given: Time for A = (x + 5) days, Time for B = (x + 2) days. Efficiency A:B = 4:5.

Formula Used: Efficiency × Time = Total Work

⇒ 4 × (x + 5) = 5 × (x + 2)

⇒ 4x + 20 = 5x + 10

⇒ x = 10

⇒ Days for A = 10 + 5 = 15 days; Days for B = 10 + 2 = 12 days

⇒ 2-day work (B + A) = 5 + 4 = 9 units

⇒ 6 cycles (12 days) work = 9 × 6 = 54 units

⇒ Remaining work = 60 − 54 = 6 units

⇒ Day 13: B does 5 units. Remaining = 1 unit

⇒ Day 14: A takes 1/4 = 0.25 days

⇒ Total days = 12 + 1 + 0.25 = 13.25 days

∴ The correct answer is 13.25 days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work. If efficiency ratio is a:b, then time ratio is b:a.

Alternate Day Work

In alternate day problems, calculate the work done in one complete cycle (sum of efficiencies of all participants) and then find the number of full cycles possible.

Pipes and Cisterns

Similar to time and work, but outlet pipes have negative efficiency. Total capacity is treated as Total Work.

Shortcut Trick

Ratio of Profit to Loss is 3 : 4 since 33.33% = 1/3 (Loss = 3 + 1 = 4 units if Profit = 3 units).

Total difference in Selling Prices = 900 − 600 = 300.

Divide 300 in ratio 3 : 4 ⇒ 7 units = 300 ⇒ 1 unit = 300/7 (approximate calculation needed).

Alternatively, use the equation: 4 × (900 − CP) = 3 × (CP − 600) ⇒ 3600 − 4CP = 3CP − 1800 ⇒ 7CP = 5400 ⇒ CP = 5400/7.

Required SP = CP × (1 + 1/6) = (5400/7) × (7/6) = 900.

∴ The correct answer is Rs 900.

Alternate Method

Given → SP₁ = 600 (Loss), SP₂ = 900 (Profit), Loss = Profit + 33.33% of Profit, Required Profit = 16.66%.
Formulas: Profit = SP − CP | Loss = CP − SP | SP = CP × (100 + P%)/100.

⇒ Let Profit = P. Then Loss = P + (1/3)P = 4P/3.
⇒ CP − 600 = 4P/3 and 900 − CP = P.
⇒ Add the two equations: (900 − CP) + (CP − 600) = P + 4P/3.
⇒ 300 = 7P/3 ⇒ P = 900/7.
⇒ CP = 900 − 900/7 = (6300 − 900)/7 = 5400/7.
⇒ Required Profit % = 16.66% = 1/6.
⇒ New SP = CP × (1 + 1/6) = (5400/7) × (7/6) = 900.

∴ The correct answer is Rs 900.

Additional Information

Percentage to Fraction Conversion

Significant percentages: 33.33% = 1/3, 16.66% = 1/6, 14.28% = 1/7, 11.11% = 1/9.

Cost Price Calculation

When profit and loss are given for two different SPs, the difference between SPs is the sum of profit and loss values.

Shortcut Trick

Water = 12.5% = 1/8 part; Milk = 7/8 parts.

Difference (7 − 1) = 6 units → 240 liters ⇒ 1 unit = 40 liters.

Initial Water = 40 liters, Initial Milk = 280 liters.

New Water = 40 + 40 = 80 liters; New Total = (280 + 20) + 80 = 380 liters.

Approximate Water % = (80 ÷ 380) × 100 ≈ 21%.

∴ The correct answer is 21%.

Alternate Method

Given: Water % = 12.5%, Milk % = 100% − 12.5% = 87.5%.

⇒ Ratio of Milk to Water = 87.5 : 12.5 = 7 :

1. ⇒ Difference in units = 7 − 1 = 6 units.

⇒ 6 units = 240 liters ⇒ 1 unit = 40 liters.

⇒ Initial Water = 1 × 40 = 40 liters.

⇒ Initial Milk = 7 × 40 = 280 liters.

⇒ New Water = 40 + 40 = 80 liters.

⇒ New Milk = 280 + 20 = 300 liters.

⇒ New Total Mixture = 80 + 300 = 380 liters.

⇒ Water % = (80 ÷ 380) × 100 = 800 ÷ 38 ≈ 21.05%.

∴ The correct answer is 21%.

Additional Information

Fraction to Percentage Conversion

Common fractions like 1/8 = 12.5%, 1/4 = 25%, and 1/6 = 16.66% help in solving mixture problems rapidly without long divisions.

Mixture Concentration

The concentration of a component is calculated as (Quantity of Component ÷ Total Quantity of Mixture) × 100.

Alligation Rule

When two mixtures or components are added, the final ratio can be found using the formula: (Quantity 1 ÷ Quantity 2) = (Rate 2 − Mean Rate) ÷ (Mean Rate − Rate 1).

Common Calculation –

Total number of students in class A is 140.

Difference between number of boys and girls is 20.

Number of boys is more than number of girls.

So, Number of boys in class A is [140 +20]/2= 80

Number of girls in class A is [140 -20]/2 = 60

Value of others class also calculate same way which is given in the table.

Calculation

Total girls = 60 + 50 + 50 + 60 + 50 = 270

Girls with scholarship = 20% of 270 = 54

Boys with scholarship = 54 + 11.11% of 54 = 54 + 6 = 60

Total = 54 + 60 = 114

Common Calculation –

Total number of students in class A is 140.

Difference between number of boys and girls is 20.

Number of boys is more than number of girls.

So, Number of boys in class A is [140 +20]/2= 80

Number of girls in class A is [140 -20]/2 = 60

Value of others class also calculate same way which is given in the table.

Calculation

A: 80 / (80+60) = 80/140 ≈ 57.14% ❌

B: 70 / 120 ≈ 58.33% ❌

C: 80 / 130 ≈ 61.54% ✅

D: 90 / 150 = 60% ❌

E: 60 / 110 ≈ 54.55% ❌

Only Class C.

Common Calculation –

Total number of students in class A is 140.

Difference between number of boys and girls is 20.

Number of boys is more than number of girls.

So, Number of boys in class A is [140 +20]/2= 80

Number of girls in class A is [140 -20]/2 = 60

Value of others class also calculate same way which is given in the table.

 

Calculation

Class D:

Boys present: 70% of 90 = 63

Girls present: 80% of 60 = 48

Class E:

Boys present: 70% of 60 = 42

Girls present: 80% of 50 = 40

Classes A, B, C:

Boys: 80 + 70 + 80 = 230

⇒ 90% of 230 = 207

Girls: 60 + 50 + 50 = 160

⇒ 95% of 160 = 152

Total present = 63 + 48 + 42 + 40 + 207 + 152 = 552

Common Calculation –

Total number of students in class A is 140.

Difference between number of boys and girls is 20.

Number of boys is more than number of girls.

So, Number of boys in class A is [140 +20]/2= 80

Number of girls in class A is [140 -20]/2 = 60

Value of others class also calculate same way which is given in the table.

Calculation

Class B:

Boys passed: 60% of 70 = 42

Girls passed: 40% of 50 = 20

Class C:

Boys passed: 60% of 80 = 48

Girls passed: 40% of 50 = 20

Class A, D, E:

Boys: 80 + 90 + 60 = 230 ⇒ 50% = 115

Girls: 60 + 60 + 50 = 170 ⇒ 30% = 51

Total Passed = 42 + 20 + 48 + 20 + 115 + 51 = 296

Total Students = 80 + 60 + 70 + 50 + 80 + 50 + 90 + 60 + 60 + 50 = 650

Pass Percentage = (296 / 650) × 100 = 45.54% = 45%

Common Calculation –

Total number of students in class A is 140.

Difference between number of boys and girls is 20.

Number of boys is more than number of girls.

So, Number of boys in class A is [140 +20]/2= 80

Number of girls in class A is [140 -20]/2 = 60

Value of others class also calculate same way which is given in the table.

Calculation

From Class A:

⇒ Boys = 30% of 80 = 0.3 × 80 = 24

⇒ Girls = 40% of 60 = 0.4 × 60 = 24

⇒ Total = 24 + 24 = 48

From Class D:

⇒ Boys = 30% of 90 = 0.3 × 90 = 27

⇒ Girls = 40% of 60 = 0.4 × 60 = 24

⇒ Total = 27 + 24 = 51

From Class B:

⇒ Boys = 20% of 70 = 0.2 × 70 = 14

⇒ Girls = 20% of 50 = 0.2 × 50 = 10

⇒ Total = 14 + 10 = 24

From Class C:

⇒ Boys = 20% of 80 = 0.2 × 80 = 16

⇒ Girls = 20% of 50 = 0.2 × 50 = 10

⇒ Total = 16 + 10 = 26

From Class E:

⇒ Boys = 20% of 60 = 0.2 × 60 = 12

⇒ Girls = 20% of 50 = 0.2 × 50 = 10

⇒ Total = 12 + 10 = 22

∴ Total students participated = 48 + 51 + 24 + 26 + 22 = 171

Given:

5, 9, 26, 103, 514, 3084

Calculation:

The logic follows here is:

(5 × 2) - 1 = 10 - 1 = 9

(9 × 3) - 1 = 27 - 1 = 26

(26 × 4) - 1 = 104 - 1 = 103

(103 × 5) - 1 = 515 - 1 = 514

(514 × 6) - 1 = 3084 - 1 = 3083 ≠ 3084

There should be 3083 instead of 3084.

∴ 3084 is the wrong term

Given:

1501, 1395, 1264, 1138, 987

Calculation:

∴ The wrong number is 1264.

Given:

312, 320, 311, 436, 389, 1718

Calculation:

Considering the above series, the logic should be

320 - 312 = 8 = (2)³

311 - 320 = -9 = -(3)²

436 - 311 = 125= (5)³

387 - 436 = -49 = -(7)²

1718 - 387 = 1331 = (11)³

After observing the logic we can say the wrong term is 389.

∴ The wrong term in this series is 389.

Quantity 1:

Given:

Compound Interest = Rs.1331

Time = 2 years

Rate = 20 percent

Formula:

C.I = P × [(1 + R/100)ⁿ – 1]

Here,

C.I = Compound Interest

P = Principal Amount

R = Rate of interest

n = Time in years

Calculation:

By using given formula

1331 = P × [(1 + 20/100)² – 1]

⇒ 1331 = P × [(1 + 1/5)² – 1]

⇒ 1331 = P × [(6/5)² – 1]

⇒ 1331 = P × [36/25 – 1]

⇒ 1331 = P × [11/25]

⇒ 1331 = 11/25 × P

⇒ P = (1331 × 25)/11

⇒ P = 33275/11

⇒ P = 3025

∴ The Principal amount will be Rs.3025

Quantity 2:

Given:

Simple Interest = Rs.1500

Time = 2 years

Rate = 10 percent

Formula:

S.I = (P × R × T)/100

Here,

S.I = Simple Interest

P = Principal Amount

R = Rate of interest

T = Time in years

Calculation:

By using given formula

1500 = (P × 10 × 2)/100

⇒ 1500 = (P × 20)/100

⇒ P = (1500 × 100)/20

⇒ P = 150000/20

⇒ P = 7500

∴ The Principal amount will be Rs.7500

The comparison of both the result:

Quantity 2 > Quantity 1

Quantity 1:

Given:

Selling Price (S.P) = Rs.12000

Profit = 20 percent

Formula:

Profit Percent = [(S.P – C.P)/C.P] × 100

Here,

S.P = Selling Price

Calculation:

We know that –

Profit Percent = [(S.P – C.P)/C.P] × 100   ……… (1)

Now,

Put all the given Values in equation (1) then we get

20 = [(12000 – C.P)/C.P] × 100

⇒ 20 = (1200000 – 100C.P)/C.P

⇒ 20C.P = 1200000 – 100C.P

⇒ 20C.P + 100C.P = 1200000

⇒ 120C.P = 1200000

⇒ C.P = 1200000/120

⇒ C.P = 10000

∴ The Cost price will be Rs.10000

Quantity 2:

Given:

Selling Price (S.P) = Rs.18000

Discount = 25 percent

Formula:

Discount Percent = [(M.P – S.P)/M.P] × 100

Here,

S.P = Selling price

Calculation:

We know that –

Discount Percent = [(M.P – S.P)/M.P] × 100 ……….

(1) Now,

Put all the given values in equation (1) then we get

25 = [(M.P – 18000)/M.P] × 100

⇒ 25 = (100M.P – 1800000)/M.P

⇒ 25M.P = 100M.P – 1800000

⇒ 25M.P – 100M.P = – 1800000

⇒ – 75M.P = – 1800000

⇒ M.P = 1800000/75

⇒ M.P = 24000

∴ The Marked Price will be Rs.24000

Comparison of both the result:

∴ Quantity 2 > Quantity 1

General Solution:

Given:

A = 35%

B = 15%

C = X%

D = 30%

Total % = 100%

So:

X = 100 − (35 + 15 + 30) = 20%

C = 20%

Now, let's compute the actual number of goods manufactured and sold by each company.

Total goods manufactured = 200

 

 

Calculations:

Let's calculate the number of unsold items for each company using the table:

 

Total unsold goods = 21 + 6 + 16 + 6 = 49.

Hence, the correct answer is 49.

General Solution:

Given:

A = 35%

B = 15%

C = X%

D = 30%

Total % = 100%

So:

X = 100 − (35 + 15 + 30) = 20%

C = 20%

Now, let's compute the actual number of goods manufactured and sold by each company.

Total goods manufactured = 200

 

 

Calculations:

Total goods manufactured by C

From earlier:

C manufactures 20% of 200 = 40 goods.

C sells 60% of 40 = 24 goods.

Unsold = 40 − 24 = 16

Defective : Non-defective = 1 : 3

This is a total of 1 + 3 = 4 parts.

Each part = 16 ÷ 4 = 4

Non-defective = 3 parts = 3 × 4 = 12

12 non-defective unsold items from shop C.

Hence, the correct answer is 12.

General Solution:

Given:

A = 35%

B = 15%

C = X%

D = 30%

Total % = 100%

So:

X = 100 − (35 + 15 + 30) = 20%

C = 20%

Now, let's compute the actual number of goods manufactured and sold by each company.

Total goods manufactured = 200

 

 

Calculations:

From the earlier table:

Goods manufactured by A = 70

% Sold by A = 70%

So, Sold by A = 70% of 70 = 49

Thus, Unsold by A = 70 − 49 = 21

Goods sold by D = 54

Required Percentage = (Unsold items from A / Sold items from D) × 100 = (21 / 54) × 100 = 38.89% ≈ 39%

Hence, the correct answer is 39%.

General Solution:

Given:

A = 35%

B = 15%

C = X%

D = 30%

Total % = 100%

So:

X = 100 − (35 + 15 + 30) = 20%

C = 20%

Now, let's compute the actual number of goods manufactured and sold by each company.

Total goods manufactured = 200

 

 

Calculations:

From the table:

Shop B sold = 24

Shop C sold = 24

Shop D sold = 54

Average goods sold by B, C, D = (24 + 24 + 54) / 3 = 102 / 3 = 34

Goods Manufactured by Shop A

= 35% of 200 = 0.35 × 200 = 70

70 − 34 = 36

Thus, the average number of goods sold by shops B, C, and D is 36 less than the total number of goods manufactured by shop A.

Shortcut Trick

Curved Surface Area (CSA) = 2πrh = 6160

Volume = πr²h = (2πrh × r) ÷ 2 = (CSA × r) ÷ 2

First, find radius (r): r = 6160 ÷ (2 × 22/7 × 35) = 6160 ÷ 220 = 28 m

Volume = (6160 × 28) ÷ 2 = 6160 × 14 = 86240 m³

∴ The correct answer is 86240 m³.

Alternate Method

Given: Curved Surface Area (CSA) = 6160 m², Height (h) = 35 m

Formula Used: CSA = 2πrh; Volume = πr²h

⇒ 2 × (22/7) × r × 35 = 6160

⇒ 2 × 22 × r × 5 = 6160

⇒ 220 × r = 6160

⇒ r = 6160 ÷ 220 = 28 m

⇒ Volume = πr²h = (22/7) × 28 × 28 × 35

⇒ Volume = 22 × 4 × 28 × 35

⇒ Volume = 88 × 980 = 86240 m³

∴ The correct answer is 86240 m³.

Additional Information

Total Surface Area of Cylinder

It includes the curved area and the area of two circular bases: TSA = 2πr(h + r).

Volume of a Hollow Cylinder

Calculated as the difference between outer and inner volumes: V = π(R² − r²)h.

Divisibility Rule of 11

Since volume involves π (22/7), the answer is usually a multiple of 11. Sum of digits at odd places − sum of digits at even places = 0 or multiple of 11.

Shortcut Trick

Let each installment in CI be Rs. 2300 (since total paid in two equal installments is Rs. 4600 → each installment = Rs. 2300).

At 30% interest, present value of installments: P = [I ÷ (1 + r/100)] + [I ÷ (1 + r/100)²]

⇒ P = [2300 ÷ 1.3] + [2300 ÷ 1.69] ≈ 1769.23 + 1360.95 = Rs. 3130.

Under SI, total amount paid for 2 years: A = P + (P × R × T)/100 ⇒ A = 3130 + (3130 × 30 × 2)/100 = 3130 + 1878 = Rs. 5008.

Wait, let us recalculate the exact values: Let P be the principal. Installment = Rs. 2300. P = 2300/1.3 + 2300/1.69 = (2300 × 130)/169 + 230000/169 = 529000/169 = Rs. 3130.18.

Actual SI total amount paid = P + P × 0.6 = 1.6 × P = 1.6 × (529000/169) ≈ Rs. 5008.28.

Difference in amount paid = 5008.28 − 4600 ≈ Rs. 408? No, let us re-evaluate. Total paid in two equal installments is Rs. 4600 ⇒ each installment = Rs. 2300. Or is the total paid = 2 × 4600 = 9200? If each installment is Rs. 4600, total paid = Rs. 9200. Let's test this: if installment = Rs. 4600, P = 4600/1.3 + 4600/1.69 = (4600 × 1.3 + 4600)/1.69 = (5980 + 4600)/1.69 = 10580/1.69 = Rs. 6260.35. Then SI amount = 6260.35 × 1.6 = 10016.56. Diff = 10016.56 − 9200 = 816.56.

What if the total principal X borrowed is Rs. 4600? 'X' pays Rs. 4600 to 'Y' in two equal annual installments ⇒ total paid = Rs. 4600, so each installment = Rs. 2300. Let's find exact principal P = (2300 × 1.3 + 2300)/1.69 = 5290/1.69 = Rs. 3130.18. No, if P is Rs. 3130.18, SI amount is P + P × 0.6 = 1.6 × P. Let's write the ratio: P = I × (13/10 + 169/100) is wrong. P = I × (10/13 + 100/169) = I × 230/169. For I = 2300, P = 2300 × 230/169 = 529000/169 ≈ 3130.18. SI interest = P × 60% = 3130.18 × 0.6 = 1878.11. Total SI amount = 3130.18 + 1878.11 = 5008.29. Diff = 5008.29 − 4600 = 408.29.

Let's check if each installment = Rs. 4600? If so, total paid in CI = 2 × 4600 = Rs. 9200. Principal P = 4600 × 230/169 ≈ Rs. 6260.35. SI interest = 6260.35 × 0.6 = Rs. 3756.21. Total SI amount = 6260.35 + 3756.21 = Rs. 10016.56. Diff = 10016.56 − 9200 = Rs. 816.56.

Wait, let's assume P = Rs. 4600 (the principal borrowed is Rs. 4600, and X pays it in two equal installments). If P = Rs. 4600, then CI installment I: 4600 = I × 230/169 ⇒ I = 4600 × 169/230 = 20 × 169 = Rs. 3380. Total paid in CI = 2 × I = 2 × 3380 = Rs. 6760. Total paid in SI = P + P × 30% × 2 = 4600 + 4600 × 0.6 = 4600 + 2760 = Rs. 7360. Difference between SI amount and CI amount = 7360 − 6760 = Rs. 600.

What if the simple interest is calculated on each installment? Or what if 'X' will pay the same principal at the same rate of simple interest? Yes! If X borrows Rs. 4600 and pays it in 2 equal annual installments of SI? If the principal is P = Rs. 4600: Under CI, installment I = Rs. 3380, total CI amount = Rs. 6760. Under SI, two equal annual installments of x: P = x/(1+rt) + x/(1+rt) is not used. In SI installment, Principal P = 2x + [x × R × 1]/100 ⇒ 4600 = 2x + 0.3x = 2.3x ⇒ x = Rs. 2000. Total paid in SI = 2 × 2000 = Rs. 4000? No, that is less than principal. Under SI, the formula for installment is: Principal + Interest = Total Installment payments + Interest on installments. P + (P × R × T)/100 = n × x + [x × R × n(n-1)]/200 ⇒ 4600 × 1.6 = 2x + x × 30 × 1 / 100 ⇒ 7360 = 2.3x ⇒ x = Rs. 3200. Total SI amount paid = 2 × 3200 = Rs. 6400. Difference between total amount paid in CI and SI = 6760 − 6400 = Rs. 360.

∴ The correct answer is Rs. 360.

Alternate Method

Given: Principal (P) = Rs. 4600, Rate (R) = 30% per annum, Time (T) = 2 years.

Formula Used:

For Compound Interest (2 equal annual installments of Rs. I_CI):

⇒ P = [I_CI ÷ (1 + R/100)] + [I_CI ÷ (1 + R/100)²]

For Simple Interest (2 equal annual installments of Rs. I_SI):

⇒ P + (P × R × T)/100 = 2 × I_SI + [I_SI × R × 1]/100

Calculations:

⇒ For CI: 4600 = I_CI × [10/13 + 100/169]

⇒ 4600 = I_CI × [230/169]

⇒ I_CI = (4600 × 169) ÷ 230 = 20 × 169 = Rs. 3380

⇒ Total amount paid in CI = 2 × 3380 = Rs. 6760

⇒ For SI: 4600 + (4600 × 30 × 2)/100 = 2 × I_SI + (I_SI × 30 × 1)/100

⇒ 4600 + 2760 = 2 × I_SI + 0.3 × I_SI

⇒ 7360 = 2.3 × I_SI

⇒ I_SI = 7360 ÷ 2.3 = Rs. 3200

⇒ Total amount paid in SI = 2 × 3200 = Rs. 6400

⇒ Difference in amounts paid = 6760 − 6400 = Rs. 360

∴ The correct answer is Rs. 360.

Additional Information

Compound Interest Installments

Each installment is treated as the present value of a future payment, compounded annually: P = I / (1 + r)¹ + I / (1 + r)² + ...

Simple Interest Installments

The total debt equals the sum of the installments plus the simple interest earned on each installment from the time of payment until the end of the loan period.